Theory of contact for geometric continuity of parametric curves

This paper develops a theory of contact for piecewise parametric curves based on the differential geometry of evolutes, polar curves, and binormal indicatrices. This theory is completely geometric, independent of parametrization and generalizes to any order. Two sets of dimensionless, characteristic numbers describing the local geometry of a curve up tonth order are defined. These characteristic numbers can be used to describe conditions for higher order contacts in an algebraic fashion. The same characteristic numbers can also be used to interpret contact conditions of up tonth order in terms of the geometry of higher evolutes and binormal indicatrices. The resulting geometric contact conditions are used to design piecewise parametric curves for Computer Aided Geometric Design (CAGD) applications.